[quoteMoreover, will doing an MS without a BA in engineering make it harder for me to find good jobs in the US
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There are no guarantees that you will be employed as an engineer in the US.
As a non-citizen engineer, most engineering positions require US security clearances; many engineering firms are required, by the US government, to hire US citizens first and must state that there are no US candidates before even considering a non-citizen candidate. Your job prospects in the US would be limited by immigration policies.