Is this representation an accurate one for the Nash Equilibrium?
$$\{\nexists s_1^{'} \in S_1 : u_1(s_1^{'},s_2) \geq u_1(s_1,s_2)\} \ \wedge\ \{\nexists s_2^{'} \in S_2 : u_2(s_1,s_2^{'}) \geq u_2(s_1,s_2)\}$$
In this case minor $s$ represents Nash Equilibrium strategies for both players and $s^{'}$ represents the other possible strategies for any player.
No, because of the 'greater or equal' comparison. Imagine that you have two strategies, $s_1$ and $t_1$, that are best responses to $s_2$ for player 1. Then your set condition won't work, as $u_1(t_1,s_2) \geq u_1(s_1,s_2)$.