Programs that are designed to interact with a human are different from programs that are designed to interact with other programs. Your square.py script is closer to the former category. Possible issues:
- the prompt ends in the middle of a line that forces a parent script to read one byte at a time instead of full lines or known chunks to avoid blocking
- child's answers are not flushed explicitly. It means that the block buffering may be enabled in non-interactive mode e.g., when it is run via subprocesswithoutpty
Here's how you could interact with it using subprocess module in its current form:
#!/usr/bin/env python
from __future__ import print_function
import sys
from itertools import cycle
from subprocess import Popen, PIPE
from textwrap import dedent
# start child process
p = Popen([sys.executable or 'python', '-u', '-c', dedent("""
        for i in range(10):
            x = int(input('Enter x-dimension: '))
            print(x*x)
        """)], stdin=PIPE, stdout=PIPE, universal_newlines=True, bufsize=1)
for n in cycle([3, 1, 4, 15, 926]): # infinite loop
    while p.poll() is None: # while the subprocess is running
        # send input to the child
        print(n, file=p.stdin)
        # read & parse answer
        data = p.stdout.readline().rpartition(' ')[2]
        if not data: # EOF
            answer = None
            break # exit inner loop
        answer = int(data)
        if answer == 1: # show example when input depends on output
            n += 1
        else: # done with given `n`
            break # exit inner loop
    else: # subprocess ended
        break # exit outer loop
    if answer is not None:
        print("Input %3d Output %6d" % (n, answer))
p.communicate() # close pipes, wait for the child to terminate
And here's the same thing but using pexpect (for comparison):
#!/usr/bin/env python
import sys
from itertools import cycle
from textwrap import dedent
import pexpect
child = pexpect.spawnu(sys.executable or 'python', ['-c', dedent("""
            for i in range(10):
                x = int(input('Enter x-dimension: '))
                print(x*x)
            """)])
for n in cycle([3, 1, 4, 15, 926]):
    while True:
        i = child.expect([pexpect.EOF, u'x-dimension:'])
        if i == 0: # EOF
            answer = None
            child.close()
            sys.exit()
        elif i == 1: # child waits for input
            child.sendline(str(n))
            child.expect(u'\\n\\d+\\s')
            answer = int(child.after)
            if answer == 1:
                n += 1
            else:
                break
        else:
            assert 0
    else: # child terminated
        break
    if answer is not None:
        print("Input %3d Output %6d" % (n, answer))
Both scripts are written to support Python 2 and Python 3 from the same source.
Note: there is -u argument in the subprocess-based script that allows to read the lines as soon as they are available even in non-interactive mode. pexpect-based script works without such switch. stdio-based programs can be unbuffered/make line-buffered using stdbuf, unbuffer utilities or by providing a pty.
You can see that even the simplest child script (square.py) requires to overcome several issues to even work at all.
Everything is simpler when the child program expects to be run from another program while remaining human-readable (debuggable). In this case, square.py could look like:
#!/usr/bin/env python
import sys
import time
for line in iter(sys.stdin.readline, ''): # get line as soon as it is available
    print(int(line)**2) # find square
    sys.stdout.flush()  # make the answer available immediately
    time.sleep(.5) # a delay to show that the answer is available immediately
It could be used from a subprocess-based module in "all at once" mode:
import sys
from subprocess import Popen, PIPE
L = [2, 7, 1] # numbers to be squared
p = Popen([sys.executable or 'python', 'square.py'], stdin=PIPE, stdout=PIPE,
          universal_newlines=True, bufsize=-1)
answers = map(int, p.communicate("\n".join(map(str, L)))[0].splitlines())
Or one number at a time:
#!/usr/bin/env python
import sys
from subprocess import Popen, PIPE
answers = []
p = Popen([sys.executable or 'python', 'square.py'], stdin=PIPE, stdout=PIPE,
          bufsize=1)
for c in [b'2', b'7', b'1']:
    p.stdin.write(c + b'\n')
    p.stdin.flush()
    answers.append(int(p.stdout.readline()))
    print(answers)
p.communicate() # close pipes, wait for child to finish
print(answers)
To get an array from your C program; you could json module:
import json
from subprocess import Popen, PIPE
p = Popen(['./c-program', 'other', 'args'], stdin=PIPE, stdout=PIPE, bufsize=1)
p.stdin.write(json.dumps({'parameter': 8}).encode() + b'\n') # send input
p.stdin.flush()
result = json.loads(p.stdout.readline().decode()) # e.g., {"result": [0, 0, 7]}
# ...
p.communicate() # close pipes, wait for child to finish