Should I say either 'I am born on Tuesday' or 'I am born on a Tuesday'? Which one is correct?
Both are grammatically correct and either can be used, but should be in the past tense.
Google ngrams suggests that both are used.
Kwabena [is the] (soul-)name of a male born on a Tuesday.
Certainly one born on Tuesday could not do better than to unite with one born on Wednesday. Expect sparks to fly from this!
With a specific date, don't use "a":
He was born on Tuesday the 2nd March 1852
Many British native speakers would interpret "I was born on Tuesday" to mean "I was born last Tuesday", so "on a [day]" would avoid ambiguity.