Yes, and ordinarily this would be the case. The use of 'no' in your example is expressive/theatrical and it adds to the meaning. When we use 'no' this way we do so to say that we really feel the negative to be true, even if it's not literally true. Take Phil Collins song 'You're no son of mine' The relationship was that of a father and son- the son's behaviour caused the father to denounce his son and proclaim he was no longer his son - of course biologically not possible, but emotionally and metaphorically, the intent was to disown/disinherit. Joseph Mengeler was no Doctor - he was a qualified MD, but his actions in WW2 broke the Hippocratic Oath and he certainly wasn't acting in the interests of human health. Donald Trump is no diplomat - he's POTUS, but lacks the delicate skills of diplomacy required of the position. Enjoy :)

The meaning is essentially the same, but the grammar is different. The second one is more usual in today’s English. They are no beggars: “No” is an adjective that modifies the word “beggars”. They are not beggars: “Not” is an adverb that modifies the verb “are”.

1. We may be poor, but we are not beggars is the same meaning as the sentence using no. 2. We are not students is grammatically correct. We are no students is something people say but not correct grammatically when written unless it is someone's speech.

For your first sentence, they are interchangeable. Honestly, I would simply stick to using "not" if you are unsure about which one to use.