How many SMD 5630 LEDs can I power with 12V 1A power supply?

IF the picture on the Amazon page is accurate to what you have, each segment has 2 27Ω resistors in series with 3 leds.

Using Ohm's law I = V/R to find out the current, Assuming 3V to 3.6V Forward Voltage drop on each led, that means the current at 12V is between

(12V Source Voltage - (3 Leds * 3.0V)) / 54 Ohms =  0.055 Amps (55 mA)

and

(12V Source Voltage - (3 Leds * 3.6V)) / 54 Ohms =  0.022 Amps (22 mA)

Since current is shared in series, this means each chip on that strip is only 22 to 55 mA.

So they are no where near your believed 150mA or 1/2 Watt.

To get the correct current, as far as the leds you have go, cut a single segment off, and using a ammeter or multimeter in current mode, measure the amperage of that single segment. That will tell you what it is using on the strip. Then I would adjust at least 10% down for a longer life. If you can't measure current, then measure the voltage across one of the resistors. Using Ohm's Law as above, take that voltage and divide by that resistance for the current flowing through it. That's what the LED should be using.

Update: It was suggested that the resistors are in parallel, and that the normal 5630 leds take 150 mA @ 3.4V. That may be true.

(12V Source Voltage - (3 Leds * 3.4V)) / 13.5 Ohms =  0.150 Amps (150 mA)

Only way to be sure would be to trace the circuit and measure the current or voltages with a multimeter.

Adding to Passerby's answer: Replicate what's on the strip. It wastes the least current. The strip is made up of repeating circuits as shown below.

^{simulate this circuit – Schematic created using CircuitLab}

You should do the same. Polarity of LEDs matters. The resistors can go anywhere in the series circuit - they're probably evenly spaced on the strip to spread the heat.