"Can OSPF be reconfigured so that all links are used? i.e some traffic via the 1000mb links and some via the 100mb link? In affect, unequal cost routing?"

Yes and no.

You can configure link cost such that OSPF will see both links as equal, then OSPF will attempt a 50/50 flow split. (If running TCP, you should see higher bandwidth utilization peeks on the gig paths, but overall utilization should average 50/50.)

Or, you could configure multiple tunnels over the gig path, and route between them and the 100 Mbps path. You'll likely run out of maximum paths (6?) and still not achieve 10:1 ratio. If maximum paths is 6, should obtain 5:1 split.

Or, depending on your equipment, you might be able to run OER/PfR and have it dynamically load balance across the two paths in proportion to their bandwidths. Neither OER/PfR directly (yet) support OSPF (or EIGRP), but they can inject statics into OSPF.

"Is this an OSPF feature or would we need to use EIGRP as our routing protocol? "

OSPF, no, EIGRP yes, atlhough don't disregard Paolo's warning about it being uncommon configuration with EIGRP. From past posts on the subject, I recall the concern is whether EIGRP can do it well especially at high speed.

With such a bandwidth difference, I do agree with Paolo to just use the gig path and retain 100 Mbps path for failure. Yes, it seems wasteful to leave the 100 Mbps path idle, but it only adds 10% to your total bandwidth, and unless the gig path is very busy (and if it were you would want more than 100 Mbps) using it will likely only slow traffic on that path.