Using resonant soft x-ray techniques we follow the magnetic behavior of a strained epitaxial film of CoCr₂O₄, a type-II multiferroic. The film is [110] oriented, such that both the ferroelectric and ferromagnetic moments can coexist in-plane. X-ray magnetic circular dichroism (XMCD) is used in scattering and in transmission modes to probe the magnetization of Co and Cr separately. The transmission measurements utilized x-ray excited optical luminescence from the substrate. Resonant soft x-ray diffraction (RXD) was used to study the magnetic order of the low temperature phase. The XMCD signals of Co and Cr appear at the same ordering temperature T_C ≈ 90K, and are always opposite in sign. The coercive field of the Co and of Cr moments is the same, and is approximately two orders of magnitude higher than in bulk. Through sum rules analysis an enlarged CO²⁺ orbital moment (m_L) is found, which can explain this hardening. The RXD signal of the (q q 0) reflection appears below T_S, the same ordering temperature as the conical magnetic structure in bulk, indicating that this phase remains multiferroic under strain. To describe the azimuthal dependence of this reflection, a slight modification is required to the spin model proposed by the conventional Lyons-Kaplan-Dwight-Menyuk theory for magnetic spinels.