10s might solve your problem but, there could be other issues as well. How old are your strings?That could be your problem. If they are more than a month old, try changing them and see if that helps.

As to squeezing the notes out of tune... My Parker (with 09s) can be pushed out of tune if I squeeze the neck too tight. Not easily, but it can be done. I think it depends a lot on how high your frets are. I don't have a mechinic's finger strength and I can do it. I suspect that anybody with strong hands could do so if their frets were high enough.

If new strings don't do the trick, maybe you could benifit with seeing if you can play with a lighter touch. If you can, does the problem go away? I do think that a good part of how people learn to play guitar is learning how to apply only enough pressure to sound a clear note. Any more pressure is just wasted energy. Experiment with using the most gentle touch you can manage. If you have a strong fingers that could be part of your problem.

On the other hand many people play with 10s or greater strings and love it. They swear they get better tone. Myself, I enjoy the light touch and bending ablity of 9s.

If all other considerations are equal, it really comes down to your personal preferances. Sorry if this is not a yes or no answer, but it might give you some other things to consider as you make your choices.