He be calling up all the time... and others

Aspectual "be" to refer to habitual action is a feature of African American Vernacular English, and it's grammatical in that context. Standard English "He calls me" or "He is calling me" are ambiguous in that they can refer to habitual actions or to one single event, but in AAVE, "he be calling me" is not ambiguous in this way AFAIK. Perhaps that is what he meant.

I wouldn't say it's more "correct." It's not Standard English. Speakers of African American Vernacular English (and just about every other non-standard dialect) have been accused of being "lazy" in the past, when in reality the language has fine shades of meaning that in some cases don't show up in Standard English. One example is than in African American Vernacular, there are three main present tenses, whereas in Standard English, there are two.

Standard
He calls me up all the time.
He is calling me up all the time.

AAVE
He call me up all the time.
He calling me up all the time.
He be calling me up all the time.

This might be what your professor was referring to. It's not more "correct," but it's more complex and provides finer shades of meaning than Standard English.

Which is more correct? It literally depends on who you're talking to.

According to my studies:

"He calls me up all the time," refers to someone calling you the name up, all the time. The verb placement and pronoun usage make this statement ambiguous.

"He's always calling me," refers to someone continuously calling you. This is incorrect because the adjective is used as an adverb, which makes the statement confusing.

"He be calling me up all the time," refers to someone calling you all the time (go figure). However, this statement is incorrect because the verb isn't conjugated properly.

Like your professor said, it's more grammatically correct, but not totally.

Chanel,

How is "He's always calling me" incorrect or confusing? "always" is an adverb: I'm always late. He's always smiling.

"He be calling me up all the time" is ungrammatical in standard English, but it is grammatical in AAVE, as Ron explained.

"He calls me up all the time" need not mean that someone is naming you "up". "Up" in this context is a particle, attached to the verb "calls", which modifies the meaning of the verb. It's just like the difference between "roll" and "roll up" in contexts such as "roll up the window" -- not to be confused with "up" used as a preposition, like "roll up the hill". A particle modifies the verb, while a preposition modifies (attaches) the following noun phrase.