Three things "rule" depth of field: lens aperture, lens focal length and distance to the object. Everything else is irrelevant, as far as DoF itself is concerned. Sensor and pixel site size aren't direct factors. The 7DII actually renders exactly the same DoF as your 5DIII (...see below).
However, once you are trying to "get the shot", other indirect factors such as shutter speed, lighting, ISO, diffraction, etc. start to come into play.
Smaller sensor cameras
seem to have greater depth of field. They don't actually... If you shoot from the same distance with the same focal length and aperture, DoF will be exactly the same.
But, we don't do that. In order to frame a subject the same way with different sensor formats, we change focal length and/or our distance to the subject. In fact, in order to get that same shot with your 5DIII (assuming you were zoomed to 400mm on the 7DII) you would have had to use a 640mm lens. Or you would have had to move closer, say to about 7 foot distance. Or a little bit of both. And doing either of these would, in fact,
decrease DoF. So, even though DoF is basically independent of sensor/film size, for all practical purposes due to the way we use them, smaller formats will render
greater DoF. (Actually there's another factor that's related to sensor size and density, called the "circle of confusion"... which also relates to the size of the image as it will be viewed and even the distance from which it's viewed... but let's just put that aside for now, so as to not confuse the issue more than necessary!)
Here you're using a long focal length quite close with the lens aperture nearly wide open, so DoF is quite shallow. Using a DoF calculator, you'll find that it's .04 foot... or about 1/2 inch! (Here's an online calculator:
http://www.dofmaster.com/dofjs.html).
Now, if you'd used a shutter speed half as fast (1/620), you could have closed the aperture down another stop (f9) to increase DoF a little bit... To .06 foot or roughly 3/4 inch. Still not very much to work with!
Increasing ISO a stop too, would have allowed you to use an even smaller aperture of f12.7, rendering a little more DoF... about .08 feet or roughly 1 inch. Better DoF, however you may not want to use such a high ISO and that small an aperture is going to start causing "diffraction", which will rob the image of some of it's fine detail. Like DoF, diffraction also isn't actually directly related to sensor size... But for practical purposes it is, because to create the same size final print from a full frame camera requires considerably less magnification of the image than it does from a crop sensor camera, and the less magnification that need be applied, the less obvious diffraction will be. So, diffraction can be a bigger concern with a crop sensor camera, than it is with FF.
Using a flash or other means of adding light might allow for slower shutter speed and an even smaller aperture, without having to use too high an ISO. But an even smaller aperture diffraction would then be a bigger problem. Plus a flash might be objectionable for other reasons (such as hot spots on the leaves of the tree that are closer to you... or a darker background in the distance behind the subject). And sometimes flash startles wildlife, too... Though, surprisingly, a lot will ignore the it (probably because they're accustomed to lightning). The sounds of the flash also may be a problem.
Other things that might help:
1. Look for a subject in better lighting, to allow you to increase DoF by stopping down a little more, without needing to use excessively high ISO or too slow a shutter speed. Just keep diffraction issues in mind.
2. Study up on "hyperfocal" focusing distances. Think of this as just simply making best use of what limited DoF you have with any given setup. For example, for this particular shot you might have had better results focusing on the critter's farther ear, instead of it's nose. That way the DoF that falls both nearer and farther from the point of focus might have been adequate for it's face and eyes, but encompassed more of it's body, too.
3. In post-processing, apply some sharpening to the slightly OOF areas. For example, a high pass filter sharpening method, applied carefully and selectively using a mask so that it only sharpens specific parts of the image, often works pretty well with furry and feathered critters.
4. It also sometimes might actually help to back off to take the shot... Or zoom to a shorter focal length... Or a little of both.... And then crop the image to the desired framing of the subject later in post-processing. More distance from the subject and/or a shorter focal length both will increase DoF. However, doing either or both of these, you'll have to weigh the loss of resolution against any gain in DoF.
5. Don't worry too much about it! The face and eye are the most critical point of focus anyway. Also, an image may have plenty of apparent sharpness when displayed in the final image size. It may be quite usable, though limited in how large you can print it.
A lot of people freak out reviewing their images at 100% on their computer monitors. Think about it. With most modern monitors and a camera with the resolution of the 7DII or your 5DIII, that's like making a five foot wide print and then judging it from 18" viewing distance... Well, of course it's going to look like crap!
While it's great to be able to zoom in to retouch images at 100% or more, most of us won't be making prints anywhere near that large or, even if we do, viewing them anywhere near that closely! So back off to 50% or even 33% or 25% for a much more realistic evaluation of sharpness and focus.
Finally, shooting with a set of gear over time, we learn what to expect simply through experience. Part of the "problem" here was that you were using a friend's camera and lens, stuff that you were less familiar with. It can be a long process to learn how things perform together through trial and error... So it might be very helpful to run a series of test shots and then study them, to get up to speed how things perform together. Maybe not with borrowed gear, but to ramp up with new stuff it can be helpful to shoot series of shots at different apertures, distances and focal lengths, just to get a feel for what DoF you might expect. This also can help you determine the optimal apertures of a lens, as well as the limitations of the less-than-optimal. Do test series with various ISOs, to study diffraction effects and more, too. In a sense, you're "calibrating" yourself so that there are fewer big surprises later.