Does OpenSSL support EAP-TLS privacy? It's an optional feature part of rfc5216, section 2.1.4.
OpenSSL doesn't implement any EAP algorithms at all. When you're connecting to Wi-Fi or using 802.1x for a wired network, then the supplicant that you're using may invoke OpenSSL to implement EAP-TLS, in which case it should work just fine. For example, the supplicant most commonly used on Linux is wpa-supplicant, and it does just this.
On the server side, you can indeed support EAP-TLS with OpenSSL, such as by using FreeRADIUS, but again, the necessary glue for EAP isn't implemented by OpenSSL, but by the RADIUS server.
In both cases, you can also use EAP-TTLS, which is somewhat more common, with OpenSSL with both pieces of software.
To determine whether your system or software supports EAP-TLS, you'll need to determine which implementation you're using.