Can I give my 1-year old son children's Zyrtec? He seems to be battling seasonal allergies. If so, what's the recommended dosage?

No. . Zyrtec not the good medicine for one year old child. This maybe not the allergy at all. Most likely this is teething syndrome and swelling gums and nasal mucosa because of this. Use nasal washing with normal saline 2 times a day and camomile tea for gums 4-5 times a day. Follow with pediatrician for other medications. .
Zyrtec . Using it under the age of 2 is not recommended as the safety is unknown. A dose of Benadryl at bedtime may be effective. The dose is 1 mg/kg. All the best.