Choosing the right mosfet to switch on-off a neopixel strip
You refer to a 2N2222 handling 800 mA.  However, the 2N2222 datasheet shows 600 mA as its maximum rating for Collector Current – Continuous.  Moreover, from the On Characteristics table, Collector – Emitter Saturation Voltage at 500 mA current could be as high as 1 V.  In typical circuits with high base drive it usually is less, eg about 0.2 V typical with 50 mA base drive, which is more current than a typical AVR output is rated for.  Anyhow, the Neopixels won't operate if your switch drops any significant voltage from the supply.  Using a 2N2222 might work ok, or briefly, or intermittently, or not at all  – depending on the quality of the  2N2222 and the amount of base drive.

You refer to a FQP30N06L handling 62 Amperes.  However, the FQP30N06L datasheet shows 32 A as its rating for Drain Current - Continuous (vs 128 A for Drain Current - Pulsed, which is irrelevant here).

Using a FQP30N06L at its maximum 32 A continuous drain current would require careful heatsinking and perhaps 10 V of VGS drive to achieve low RDS(on), like 27 mΩ. With low VGS of say 3.3 V, RDS(on) might be twice as much.  For example, if it were 45 mΩ, going above 4 or 5 A current would be problematic without careful heatsinking:  4 A → P = 4²·0.045 = 0.72 W, enough to heat up a not-heatsinked unit's junction by about 45℃ or 80℉.

But the FQP30N06L should work ok for your 600 mA typical-case scenario, and also should be ok with the 2 A worst-case as well.