Squeeze is a verb: e.g. : Yesterday I was very busy but I managed to squeeze in a dinner with my mom. Squeak is a noun, and verb, and sound, like the sound a mouse makes. e.g.: I think something is wrong with my car because I heard a squeak.

I am not sure how either of those words would be used in a situation like that. Squeak is a high pitched sound and squeeze in when something is pressed together. An example is I "squeezed" a tennis ball.

squeak = sound squeeze = action

You can't use 'squeak' or 'squeeze' on their own in that context. You have to use them with 'in' - and these are somewhat idiomatic uses. I find it rather odd that neither of the previous answerers are aware of this use. I just squeaked in, or I just squeezed in are, as you suggest, quite similar. 'Squeaked' would be more when you just did something in time -e.g. I had to get my application in by 5, and I only just squeaked in. 'Squeezed' would be more if you had to fit in - e.g. I only just squeezed into the packed room. It could also be used for other things, such as 'I only just squeezed mum's gift into my bag' or 'the doctor has rearranged his schedule, and he can squeeze you in at 3:15'.

I think what you're looking for is a way to use these words "in case I struggled with something and I made it." These each make phrasal verbs with that meaning: squeeze by and squeaked by. I almost didn't have enough money for rent, but in the end I managed to squeeze by. I didn't pay my rent last month, but I managed to squeak by my landlords door without him hearing me. Both can be used for describing "making it" despite adversity or trouble, but squeeze by conveys a meaning that someone is barely "making it," and squeak by implies some kind of escape as well. "Indiana Jones squeaked by the moving wall just before it closed."