Wireless is fairly strict, it's usually one client to one wireless association, even if using a client ethernet bridge inline - they usually only expect one mac address to pass (not if you have a client wifi bridge with 5 devices behind it). This is at least in the enterprise-world ala Cisco or Aruba.
I've not used one of these consume "wifi repeaters", but what I presume they're doing is some sort of either network address translation, or some mac address rewriting to make work, so all traffic looks like one host, or in essence another router. Not much different from your cell phone doing tethering over cell (unless you use apple crap, ugh).
What I'd say to do is look for one of these repeaters, or client bridges, put it as close to the signal as possible, output the ethernet jack to the WAN side of a consumer router/firewall, and wire the LAN ports back to whatever you have there, or use it's wireless to send your own SSID off the router wifi for your clients indoors. If a repeater can't pick up the signal, get one with an external antenna receptical, I'm sure someone makes one, and get the cantenna as mentioned to point at your neighbor's signal.
[neighbor wifi] -- [your wifi repeater] -- [router wan port] -- [wired/wireless clients] -- win!
The router will look like a normal single mac/client to the bridge, which should be good enough to pass over it to the neighbor wifi. This will need to NAT traffic properly to do so, but most routers/firewalls do this anyways.
I've never had to get very creative to hijack others' wifi to get others on with more than one client, without just breaking in with a single client and plundering that way, but in theory this should be sound, as long as the upstream wifi sees a 1:1 relation between client, mac-address, and traffic.
-mb