Let a connected, undirected graph $G=(V,E)$. Find from the set of all MSTs, the MST, $T$ such that $d_T(u)$ is the highest ($d_T(u)$ is the degree of $u$ in the tree $T$).
Now, I thought about modifying Kruskal's algorithm: We sort the edges but with a priority to edges which include the vertex $u$. i.e. If there are two edges with the same weight then the edge containing vertex $u$ will be the first one.
Is that a valid solution?
You are facing multi-criteria optimization. You need to decide what you want to optimise : either the weight of the tree, or the number of edges including the vertex $u$, or some combination of the two.
Your solution (or what I am getting from your description) provides, among the minimum spanning trees, the one in which has the biggest number of edges connected to $u$. This is not the same thing as the cheapest tree which holds all edges connected to $u$ (which you would get by starting Kruskal with already all edges adjacent to $u$).
To put it in other words, $\min \max \neq \max \min$
Am I clear enough?