As your source points out, there are Health concerns with some acrylic mouthguards, if they contain BPA. However, the concerns are, according to the FDA, negligible and propose no danger to you.
Ask your dentist whether the mouthguard in question does contain BPA and act accordingly.**
As required, the Schein MSDS for “Easy Flow Acrylic Powder” listed three hazardous ingredients: Dialkyl Phthalate (CAS# 84-66-2), Titanium Dioxide (CAS# 13453-67-7), and Mineral Pigments (CAS# 57453-37-5), but made no mention of BPA, the chemical of concern to the consumer.
So what is BPA?
Bisphenol A (C15H16O2), commonly abbreviated as BPA, is an organic compound with two phenol functional groups. It is a difunctional building block of several important plastics and plastic additives. With an annual production of 2–3 million metric tonnes, it is an important monomer in the production of polycarbonate.
BPA affects postnatal development of embryos, and has many negative health effects on mice.
As always, sola dosis facit venenum (the dose makes the poison) and
the Food and Drug Administration (FDA) has said that BPA is safe at the very low levels that occur in some foods. This assessment is based on review of hundreds of studies.