Because of the [no-computers] tag, I waited to post this answer until somebody else already proved optimality.
A graph-based approach to this problem is to first compute all-pairs shortest-path distances in an undirected graph with a node for each grassy square and an edge between nodes that are horizontal or vertical neighbors. Then solve a traveling salesman problem on a complete graph where the shortest-path distances are the edge weights. As in my answer to a similar problem, introduce a dummy node adjacent to all other nodes to allow the starting and ending squares to be anywhere.
As expected, the resulting optimal objective value is $110$, which corresponds to $111$ squares mowed:
