Should I trust my employer if they insist on 1099 work instead of W-2 employment?

Short Answer: Be cautious; it might be misclassification if you lack control over your work.

Full Answer: If your employer insists on classifying you as a 1099 independent contractor instead of a W-2 employee, first examine the nature of your work.
As a 1099 worker, you are subject to self-employment taxes and do not have employee benefits such as health insurance, overtime pay, and unemployment protection. You must report your income on your taxes in Schedule C of Form 1040. Additionally, you may have issues with allowable expenses related to this work, and you will pay self-employment tax on the entire income.
The IRS has guidelines based on three factors—behavioral control, financial control, and relationship type—to determine the proper classification. You may be misclassified if your employer controls your schedule and work methods or provides tools and ongoing work.
Misclassification is illegal and can result in tax liabilities and penalties for the employer. For clarity, ask your employer about the reasons for the classification, and consider consulting with the IRS or an employment attorney if you suspect misclassification.
General Federal Standard: Misclassifying employees as independent contractors violates the Fair Labor Standards Act (FLSA) and IRS regulations.

General Federal Standard: Misclassifying employees as independent contractors violates the Fair Labor Standards Act (FLSA) and IRS rules.