Odds of this happening in euchre
Assuming a Euchre deck with 24 cards (9, 10, J, Q, K, A of each suit, no joker) and perfect shuffling, the chance that the 21st card in the deck (the card that gets turned up) is a particular card is 1/24. Thus, if you play 4 deals, there is a (1/24)^4 = 1/331776 = 0.0003% that all four deals will turn up the Jack of Spades.
There's much higher chance of all four deals turning up the same card: (1/24)^3  = 1/13824 = 0.007%. That's because there's a 100% chance that the first deal turns up some card, and each subsequent deal has a 1/24 chance of turning up the same card as the previous deal.
If you play N deals over the course of the night, the probability that at some point all four deals turn over the same card consecutively is (N - 4) * (1/24)^3 (the sequence can start on any but the last 3 deals, and the probability of the sequence is (1/24)^3). This works out to about 0.1% if you play 20 hands and 0.7% if you play 100.