Min Cost Solution

Build all edges (complete graph), then run Kruskal:

function minCostConnectPoints(points):
    n = points.length
    edges = []

    for i from 0 to n-1:
        for j from i+1 to n-1:
            dist = abs(points[i][0] - points[j][0]) + abs(points[i][1] - points[j][1])
            edges.add((dist, i, j))

    sort(edges)
    uf = new UnionFind(n)
    cost = 0
    edgesUsed = 0

    for each (dist, u, v) in edges:
        if uf.union(u, v):
            cost += dist
            edgesUsed += 1
            if edgesUsed == n - 1:
                break

    return cost

Time: $O(n^2 \log n)$. There are $O(n^2)$ edges, $O(n^2 \log n)$ sort.

For large $n$, Prim's algorithm with a heap is faster on dense graphs.