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This is the Neu-Vita Occulizer, a vintage medical device for massaging your eyeballs. It's now up for auction on eBay for your quacktastic pleasure. From the auction description:
Link (Thanks, Michael-Anne Rauback!)A multipurpose eye massager from Neu-Vita, Ltd, England, marked with British patent 363101. Red rubber squeeze bulbs force a concave plunger against each eyeball when squeezed. When the thermoplastic eye cups are pressed firmly against the eye, releasing the bulbs will create a vacuum in the cup. The opposite end of the device is used to massage the eye by rotating each cylinder, thus twisting the eyelids.