Using a pull buoy exhausts my groin muscle so quickly, I can only manage 1 lap before needing rest. The whole time I am miserable. I hate it.
Do most people have this initial discomfort? Then, they ease in to using them by simply using them as much as tolerable?
It rests between your legs. You should not be squeezing it
Don’t squeeze it like a ThighMaster…relaaaaaxxxx.
They make an ankle version too.
Using a pull buoy exhausts my groin muscle so quickly, I can only manage 1 lap before needing rest. The whole time I am miserable. I hate it.
Do most people have this initial discomfort? Then, they ease in to using them by simply using them as much as tolerable?
as another poster said, they’re not meant to be squeezed.
While I don’t bother with them anymore (by luck, I have good body position and can focus on very minimal kick while working on stroke technique)… I remember in masters that on occasion a pull buoy would pop out from someone while swimming - including myself; I guess you could say that would be an indicator that they’re being squeezed properly. No big deal if that happens, it’s just a workout.
Enjoy the journey, when you “hate” something (like pull buoys) try and flip the hate into a mental challenge and growth opportunity. Triathlon is chock full of things “average” people hate, so instead they get to enjoy watching TV and turning into couch potatoes. I’ll tip my hat to you and anybody who embraces the suck - 'cause it won’t be long until it’ll feel natural (and, of course, something else will suck 
).
No not normal
Put closer to knees not in your junk.
how is your core strength? When I shift the buoy to my ankles I really feel the core being engaged. I bet you are overcompensating with your legs hugging the buoy because your core strength needs improvement.
Not normal, maybe try a band along with the pull buoy so it keeps your legs in.
Well, this is a new one!
There are different types of buoys. Some are one piece of foam, while others are two pieces of foam with some cord between them. Try the other kind. Also, you could ditch the buoy and just use a band (tie your ankles together), which would be better for your swimming anyway. 
Haven’t heard of this problem before. You could make a pull buoy that attaches to your leg (like they use in run/swim races).
You could also just go with the wetsuit “cheater” shorts (most wetsuit brands offer them).
If you’re using a very slim pull buoy you could try a slightly fatter one with a more pronounced “bulge” at either end. This should reduce to amount of squeeze you have to apply to keep it in place; as others have said, this should be pretty minimal.
At the end of the day if it slips down your legs or pops out altogether, it’s not the end of the world; it happens to me every now and then, especially on push-offs from the wall.
OP didn’t describe exactly what was happening. My first thought was, as a beginner maybe you’re “fighting” the pull buoy by trying to keep your head up too high. You should be able to relax and just float.
You shouldn’t be sqeezeing it. All it needs to do is sit there comfortably. When I do a pull set I might have to move it a tad when I hit the wall for a short rest, but otherwise I can usually get it back to my “comfort” spot during the flip turn by opening my legs a bit to let it move a tad
Rather than repeat the “don’t squeeze” advice of others, I’ll expand.
Until I retired from swimming, I’d just go without a pull buoy as I have pretty good body positioning anyway. In a wetsuit, my legs ride so high that kicking gets nuthin’ but air, so I don’t bother kicking. For me ironman was a 3800m pull set In the pool I can kick but generally it’s only for steering. I could do pull buoy sets without one and that worked fine for me and my coach.