You are asking 2 different questions.
Bochs is able to load any standard bootable disk. Without knowing how you were loading your code in dosbox, I'll just give you the options to make this happen. (described in
http://wiki.osdev.org/Bootloader)
1. Use a pre-established bootloader and have it load your kernel (Grub is one example of this.
http://wiki.osdev.org/GRUB)
2. Write your own bootloader and install it to the boot sector of a floppy image. (
http://wiki.osdev.org/Rolling_Your_Own_Bootloader)
There is a 3rd option as well. If you use QEMU, it can boot directly into any multiboot compliant application without a bootloader. While not a long term development option, it is rather beneficial in the early stages.
Your other question is about how to debug in bochs. iansjack so helpfully refered you to the bochs documentation. In there, you will find that bochs has several modes of debugging. The first mode uses a built in debugger that will pop up a window and you can interact with the memory, registers, etc. Another mode allows you to attach a gdb debugger to bochs and actually debug your application in code. The built in debugger is easier to setup and probably more relevant for the stage that you are at, while the "real" debugger is a better long term option but requires more setup with your IDE/compiler chain.
-http://wiki.osdev.org/Bochs#Bochs_debugging_facilities
-http://bochs.sourceforge.net/doc/docbook/user/internal-debugger.html