Can a non-exempt employee that works on-call on regular basis (Once a month for a whole week) deduct anything out of the ordinary? Like a computer used for work, bills?

Not sure what you mean by "non-exempt" employee.  If you are a W-2 employee, you cannot deduct any job-related expenses on your federal return.  The tax reform laws eliminated job-related expenses as a deduction on your federal tax return.

https://ttlc.intuit.com/questions/4482873-which-federal-tax-deductions-have-been-suspended-by-tax-reform

Wild guess here, but is the "statutory employee" box on your W-2 checked?