Does Hadwiger conjecture imply that NP = coNP?
I think the problem here is that you are stating the converse of Hadwiger's conjecture. The conjecture (according to Wikipedia) states that "if a graph needs at least k colors for a proper coloring, then it contains a $K_k$ minor (i.e. there exist subgraphs $G_1,\ldots,G_k$ as you state)". You are talking about the assertion that "If a graph has a $K_k$ minor, then it needs at least k colors for a proper coloring".

These two are not equivalent. Indeed, the latter statement is false. Consider for example a $K_{n,n}$. This graph contains a $K_n$ minor (take a perfect matching, this gives a collection of $n$ connected components all of which have edges to each other). However, its chromatic number is 2.