Boro, who sit bottom of the table, were due to host Collieries, who are just one place above them, on Saturday.
Games in the Northern Premier Division were given the go-ahead despite the coronavirus outbreak. But Gainsborough Trinity, who Rangers played last Tuesday, have a player self isolating.
The wife of Rangers chairman John Bromely also has an underlying health issue.
A statement released by Boro said: “We thought it correct we clarify information regarding our game with Atherton Collieries.
“With one of our Tuesday opponents from Gainsborough Trinity having to self isolate we spoke with the league and it was agreed that the safest thing was to postpone our fixture.
“With us also having a few people around the club who could be affected in such a situation, including chairman John Bromley whose wife currently has some serious health issues, the decision brought home the importance of reducing risk and protecting people at this time.
“To clarify, we currently have nobody diagnosed or showing signs of contracting coronavirus.”
Elsewhere in non-league, Kidderminster Harriers National League North clash with Leamington was also called off due to the coronavirus pandemic.