They had the USSR's nukes on the island. They were ready to fire (unbeknownst to Kennedy). Cuba had them, and the crisis had escalated so much Cuba wanted to launch. Castro demanded they be allowed to launch[1]. But Khrushchev declined. They wanted to bomb us (and unleash Armageddon) because they felt aggrieved.
Khrushchev was so taken aback by their cavalier attitude towards nukes that he took them back. So yes, for a few months they were a nuclear power.
This is confirmed by the USSR archives of that time plus interviews with Cuban officials back in the 90s where they conceded they wanted to bomb us and only Khrushchev kept it from happening.
Excerpt of above link (Khrushchev's response to Castro): "In your cable of October 27 you proposed that we be the first to carry out a nuclear strike against the enemy's territory. Naturally you understand where that would lead us. It would not be a simple strike, but the start of a thermonuclear world war."
Cuba has never been a nuclear power. Maybe check this https://en.m.wikipedia.org/wiki/Cuban_Missile_Crisis