Open the recliner fully using the hand lever. Lay a tarp down to protect your carpet or flooring. Turn the recliner upside down -- get some help -- onto the tarp so the underside is accessible.
Look over the framework of the recliner for loose screws or bolts. Tighten these using a wrench. Replace any that are rusted or otherwise damaged.
Apply mechanical lubricant to the recliner sprints and two-part hinges. Do not over-spray the lubricant or it may stain the upholstery or later drip onto your floor. Turn the recliner back over; sit in the chair and rock gently to check that the squeaking is gone.