Am I authorized to wear a deployment patch?

Authority to do... comes from a memo, directive, or AR. Ask for one of the above from the CoC and you're GtG. People will cite AR 670-1 till they are blue in the face but the "authority" needs to be memorialized in some document. Unfortunately, from my interpretation, you need all of the following not some: military orders plus CZTE, HFP and/or IDP

What does your 1SG say about it? He/she would certainly know.

Depends on where you were while on orders (or what happened).

Per ARCENT....
Shoulder Sleeve Insignia-Former Wartime Service (SSI-FWTS) is authorized for Soldiers who
received CZTE AND HFP/IDP AND were either deployed in support of Operation Inherent Resolve in Iraq, Jordan, or Syria between 15 June, 2014 and a date TBD; or were deployed in support of Operations Freedom’s Sentinel in Afghanistan, Pakistan, Yemen, or Djibouti between Jan. 1, 2015 and a date TBD. Soldiers who deployed in the CENTCOM area of operations (AO) on training exercises or in support of operations other than OFS/OIR are not authorized the SSI-FWTS, unless those exercises or operations became combat or support missions to OIR/OFS.

As LTC Giwa emphasized, you might want to check in with your CoC on this if you feel you meet the requirements.You will need this documented and put in iPerms.