A slight correction. The placement of the apostrophe matters. My neighbor's son is the son of my neighbor (singular). My neighbors' son is the son of my neighborS. But you're right. The son of my neighbors is grammatically correct but doesn't sound right on either side of the Atlantic.
dei = 'de i' miei vicini is masculine plural - all those i's. Della mia vicina. Ha a bunch of 'a' because vicina is femine. in otherwords: genitive (English 's) is expressed with de + article de il = del vicino, della vicina, de i = dei vicini, delle vicine (feminine plural, if there are 2 female parents/adults as neighbors)
This is an English oddity. "He is my neighbors' son" with the apostrophe after the "s" is the grammatically correct way to indicate the possessive of a plural in this case. This is accepted. (if the neighbor was a single parent it would be "vicina" or "vicino" not vicini.) But if you simply spoke English and never wrote it, there would be no audible difference; perhaps this explains some of the confusion by those wanting to use "neighbor's"