Linksys WRE54G, will this device work with a non Linksys Access Point&/or WiFi Router?

The Linksys WRE54G Wireless-G Range Expander, will this device work with
a non Linksys Access Point &/or WiFi Router?

I have a situation where I want to piggy back off of a non encrypted
publicly accessable free WiFi access that is provided in the shopping
area of my building.
I don't know the make of the Access Point / WiFi router.
I can get a good signal at the window of my kitchen but rest of place
has no signal, I want to add a repeater such as the WRE54G to get the
signal to the rest of my place. Will this work?

If not any suggestion for a repeater that will?

Thank you.