Is MIN or MAX-True-2-XOR-SAT NP-hard?
Build an implication graph, in the style of solving $\text{2-SAT}$: each literal gets a $x_i$ node and a $\neg x_i$ node. Each clause is turned into implications, each implication gives a directed edge from the antecedent-node to the consequence-node. 
Collect all the connected components of this graph.
Each connected component will have only $2$ states: since each relationship is of the forms $x_i=x_j$ or $x_i \ne x_j$; this means if you assign even a single variable, the entire component will be force-assigned. This means you only have $2$ choices for each component.
For each connected component, greedily choose the assignment state that will increase (for $\text{MAX}$; decrease for $\text{MIN}$) the truth values in $\mathbf x$.