What is correct grammar - most proud or proudest?

Both are correct. Use proudest except where most proud seems better. Proudest is the English way of making the superlative, and most proud is the French way. Since the Norman Conquest of 1066, which imposed a form of French upon English speakers, using frenchified vocabulary and grammar may be seen as somehow higher class. Utter nonsense, but there you go. One rule that almost works is this: monosyllabic or native English adjectives take -er and -est, borrowed polysyllabic adjectives take "more" and "most."