Teaser 2840: Signs of the times

From The Sunday Times, 26th February 2017 [link]

I was taking a gentle morning drive along the straight road to Secombe that passes through Firsk. Before reaching Firsk I passed three signposts, each giving the distances to Firsk and to Secombe (to the nearest mile). All six distances displayed were different and, amazingly, all were perfect squares [less than 100].

What were the two distances given on the first signpost (furthest from Firsk)?

I have added the “less than 100” condition to reduce ambiguity in the puzzle.

[teaser2840]