Charcoal, 55 bytes
ＢＮＮＪＮＮＰψ↷⎇⁼δ/⎇∧⁻⁻θ¹ζＩε⁷Ｉ3⎇∧ＩζＩε⁵Ｉ1Ｗ¬℅ＫＫ«¹Ｆ³¿∧κ℅ＫＫ«↷κ¶↷κ
Try it online! Link is to verbose version of code. Edit: I don't know why but that code isn't working right now. Here's a tweaked version instead. Explanation of the original code:
ＢＮＮ
Draw the outer box.
ＪＮＮＰψ
Move the cursor to the entry point and make a hole in the box at this point, so this becomes the background.
↷⎇⁼δ/⎇∧⁻⁻θ¹ζＩε⁷Ｉ3⎇∧ＩζＩε⁵Ｉ1Ｐ
Rotate to the appropriate direction to print the initial line segment.
Ｗ¬℅ＫＫ«¹
While the cursor is over a hole, draw the next line segment.
Ｆ³¿∧κ℅ＫＫ«↷κ¶↷κ
Look for a hole in which to draw the next line segment. This works by repeating the same code three times. The first time, nothing happens. The second time, if the cursor is not over a hole, it rotates right 45°, then takes one step to the right, then rotates right 45°. This has the effect of reflecting it in one axis. The third time, if the cursor is still not over a hole, it rotates right 90°, then takes one step to the right, then rotates right 90°. This has the effect of changing the reflection to the other axis. If there still isn't a hole, then a corner or infinite loop has been reached, so the outer loop halts.