I'm no code expert but as I understand it, yes if you use fire caulk as per exception #2 in 713.3.1 and you are going to surface mount the switch on the drywall in the garage.
713.3 Fire-resistance-rated walls. Penetrations into or through fire walls, fire barriers, smoke barrier walls and fire partitions shall comply with Sections 713.3.1 through 713.3.3. Penetrations in smoke barrier walls shall also comply with Section 713.5.
713.3.1 Through penetrations. Through penetrations of fire-resistance-rated walls shall comply with Section 713.3.1.1 or 713.3.1.2.
Exception: Where the penetrating items are steel, ferrous or copper pipes, tubes or conduits, the annular space between the penetrating item and the fire-resistance-rated wall is permitted to be protected as follows:
In concrete or masonry walls where the penetrating item is a maximum 6-inch (152 mm) nominal diameter and the area of the opening through the wall does not exceed 144 square inches (0.0929 m2), concrete, grout or mortar is permitted where it is installed the full thickness of the wall or the thickness required to maintain the fire-resistance rating; or
The material used to fill the annular space shall prevent the passage of flame and hot gases sufficient to ignite cotton waste when subjected to ASTM E 119 or UL 263 time-temperature fire conditions under a minimum positive pressure differential of 0.01 inch (2.49 Pa) of water at the location of the penetration for the time period equivalent to the fire-resistance rating of the construction penetrated.
713.3.1.1 Fire-resistance-rated assemblies. Penetrations shall be installed as tested in an approved fire-resistance-rated assembly.
713.3.1.2 Through-penetration firestop system. Through penetrations shall be protected by an approved penetration firestop system installed as tested in accordance with ASTM E 814 or UL 1479, with a minimum positive pressure differential of 0.01 inch (2.49 Pa) of water and shall have an F rating of not less than the required fire-resistance rating of the wall penetrated.
http://publicecodes.cyberregs.com/icod/ibc/2009/icod_ibc_2009_7_sec013.htm
Side note: whats wrong with it where it is in the picture? Don't want it inside your house looking ugly? Also, some of those wires look a little short. You may want to cut as much of the BX as you can leaving just enough to connect it.
The National Electrical Code requires you to maintain 1 ¼" spacing back from the edges of framing members.
*300.4 Protection Against Physical Damage. Where sub-
ject to physical damage, conductors, raceways, and cables
shall be protected.
(A) Cables and Raceways Through Wood Members.
(1) Bored Holes. In both exposed and concealed locations,
where a cable or raceway-type wiring method is installed
through bored holes in joists, rafters, or wood members,
holes shall be bored so that the edge of the hole is not less
than 32 mm (1 ¼") from the nearest edge of the wood
member. Where this distance cannot be maintained, the
cable or raceway shall be protected from penetration by
screws or nails by a steel plate(s) or bushing(s), at least
1.6 mm (1/16" in.) thick, and of appropriate length and width
installed to cover the area of the wiring.
Exception No. 1: Steel plates shall not be required to
protect rigid metal conduit, intermediate metal conduit,
rigid nonmetallic conduit, or electrical metallic tubing.
Exception No. 2: A listed and marked steel plate less than
1.6 mm (1/16" in.) thick that provides equal or better protec-
tion against nail or screw penetration shall be permitted.
So if you drill through the center of your 2x6" rafters you are complying with the NEC.
However, if you are worried about future damage to your wiring from roofers, you could use armored cable. But if a responsible, licensed, insured roofer is hired they should use the proper length nails and there shouldn't be any threat to you cables.
If you really want to overbuild it use rigid pipe and wire and rest easy when the roofers come. 😉
Best Answer
More information details would be needed to work out a definitive answer but in general the answer is yes if you keep the hole in the middle 1/3 of the joist depth. The worst thing to do is cut the bottom 1/3 where the joists are in tension and cutting it there will severally reduced the strength and create stress concentrations that will most likely cause cracking over time.