The clock analogies when explaining Special Relativity work because the clocks we imagine (ie. the ones with ticking hands and a face, or even digital ones) are physical objects built according to physical laws. For example, a mechanical clock depends on a spring's motion, while an electric clock depends on the vibration of crystals.
All physical laws undergo "time dilation" in conceptually the same way: the spring would take "longer" to unwind the same distance, and the crystal would take "longer" to vibrate (the exact details are unimportant). These macro-scale physical effects are in fact macro-scale manifestations of time dilation effects on a quantum-mechanical level, to do with the momenta of subatomic particles. Momentum, of course, depends on time, and so is not exempt from time dilation just because the particle in question exists on a quantum scale.
Cells, molecules, etc. are all made of atoms which are made of subatomic particles. Their functioning depends on the kind of processes just mentioned, occurring at the quantum level. For example, the rate of functioning of mitochondria depends on chemical reaction rates, which in turn depend on the same quantum-level effects. Anything made up of subatomic particles will thus be subjected to time dilation in the same way, so (in identical situations) a human cell would age at the same speed as a quartz clock.