Prouder and more proud are both correct and acceptable to use. Prouder is more common as it is grammatically correct.
Prouder and more proud are called comparative adjectives in English which means they compare two things.
Prouder and more proud as comparative adjectives
When we form a comparative adjective, we use +er when the word is one syllable and more+adjective when it is two syllables or more.
Proud is a one-syllable word and therefore the correct form of proud as a comparative adjective is “prouder”
However, many native English speakers find “prouder” a little bit strange to say so for that reason they say “more proud”
I have never been prouder of you.
I couldn’t be prouder of you.
In my experience, “more proud” is more common in the U.S.A, and “prouder” is more common in the U.K.
Proudest and most proud as superlative adjectives
A superlative adjective compares one thing to all the other things in its group.
The rules are roughly the same as comparative adjectives and for that reason “the proudest” is the correct form.
This is the proudest day of my life.
I can’t imagine “most proud” in place of “proudest” in this situation.