My first question is are all these devices cisco hardware? If not you can not use EIGRP since it is cisco proprietary. Next the path that will be taken is dependent on how you configure the routing protocol you pick. Also OSPF is a link state routing protocol and EIGRP is a distance vector protocol which both have pros and cons. There is a little more information needed to answer your question about the routing protocol to use like what does your total network look like and are you running all cisco gear or mixed gear along with what is your private ip address scheme for your vlans?
EIGRP is cisco proprietary. So it is supported only by cisco devices. To use it you have to make sure all your layer 3 (routing) devices are cisco. Otherwise, OSPF should be used.
Back to your questions, If I'm not mistaken, both EIGRP and OSPF will choose the path that passes through "Core Switch 2" given that they are configured correctly. And given that the 10Gbps SM link is not disabled by STP or any thing. Since OSPF uses shortest path routing with link costs (10GBps costs less). And EIGRP also calculates routing metrics according to link "cost + delay" by default. They will probably both choose the same "Core Switch 2" path since hop count in both paths is the same. Thats as far as I know.