What to do if I want to add my child to my employer sponsored health plan but they will not let me as it is not a qualifying event?

Okay more information is needed here as to why they say this is not a qualifying event.  Although it is true that birth of a child is considered a qualifying event to ad them outside the open enrollment period, so to is divorce, separation, a court order in divorce mandating coverage, etc. Here is more:

"Under the Internal Revenue Service (IRS) rules, an employee may change their health insurance deductions during the plan year only after one of the following pre-taxed-qualifying events: You have a change in family status (e.g. marriage, birth, death, legal separation, divorce, only dependent child's attaining the maximum age for coverage up to age 26 years). You are enrolled in an HMO and no longer live or work in that HMO's service area and you must choose another HMO or the Empire Plan (PPO). Your spouse loses coverage due to termination of employment and you apply for coverage for your spouse. You first become eligible for health insurance coverage after the beginning of the tax year. Your employment terminates or you retire. Your spouse has a change in employment status which results in either acquiring or losing eligibility for health insurance coverage. You receive a divorce/legal separation and are required under a court order to provide health insurance. coverage for your eligible dependent children and/or legally separated spouse. There is a significant change in your or your spouse's health coverage which is attributable to your or your spouse's employment." 

If you fall in to any of these categories contact your state Department of Insurance and file a complaint.  That should get things shaking in your favor.  Good luck.

Unfortunately, the fact that you could face penalties does not require your employer or its insurer to let you add the child at this time. Employer-sponsored health insurance can only be added, changed, or dropped 1) during the annual open enrollment season; or 2) when there is a qualifying event. If it's not open enrollment and if your child was not just born (since birth of a child is a qualifying event), you cannot add your child at this time.