It's not in any code I know of. However, the beam, support posts, foundation (if applicable), and hangers should be designed by a structural engineer.

We have a 2 story house and the beam is for the 1st floor, does this make a difference because he said that's the reason he couldn't do it according to code?

if the beam he installed under the floor joist is sized to hold up the weight then a flush header or beam would be the same. I never have heard of a code issue in doing a properly sized flush beam. If you have plumbing running thru the joist or wiring that cannot be located then that would be different. Yes it is a lot more work to install a flush beam and joist hangers, but if this is to achieve a more open look or one large room then making it flush is the only option. Was there a building permit pulled and if there was call the inspector and ask him. A drop header in the line of site makes it a divider of rooms. Tell the contractor you will have the inspector check the code and see what the contractors says. I bet he will change his mind as to it being a code issue rather more of a time saving issue for him. I have been a carpenter contractor for 40 years so I do have a little background on this.

What do the engineer stamped drawings that were presented for the building permit show? Unfortunately this is not necessarily an internet question as all anyone can go by is hypothetical's. No ine here has been to your home, seen the plans, or spoke to the engineer.

No, you can’t do anything the engineer says can’t be done.

All the above comments are spot on. If this is a permitted job you need to adhere to the structural docs submitted, stamped, and approved. Its possible to alter if the engineer will sign off on the variation and stamp it for the governing entity. In short though flushing a beam is generally a perfect alternative to under joist application so long as the bearing points remain the same.