ENH: add a keyword to the eq method to consider NAs equal
### Code Sample

This works as expected. Comparing nan with nan evaluates to False.
```python
>>> import pandas as pd
>>> import numpy as np
>>> s1 = pd.Series(np.nan)
>>> s1.eq(s1)
0    False
dtype: boolean
```
However, this does not work as expected. Comparing nan with nan evaluates to nan.
```python
>>> import pandas as pd
>>> import numpy as np
>>> s1 = pd.Series(np.nan).astype(pd.StringDtype())
>>> s1.eq(s1)
0    <NA>
dtype: boolean
```

### Problem description
As in other programming languages I would expect the comparison between nan values for equality evaluates to False. In the first case, this works as expected. However, for the second it does not. I read the docs for the [Nullable Boolean data type](https://pandas.pydata.org/pandas-docs/stable/user_guide/boolean.html). It implements the Kleene Logic for Boolean data type. So, the second case is expected behavior, too. The issue is even explicitly stated in the docs.  It is just not very consistent between the two cases.

I searched for similar bug reports and was not sure if they are directly related. For reference please look here:
- [https://github.com/pandas-dev/pandas/issues/20442](https://github.com/pandas-dev/pandas/issues/20442)
- [https://github.com/pandas-dev/pandas/issues/36941](https://github.com/pandas-dev/pandas/issues/36941)

### Expected behavior
I would expect that the two cases behave consistently. In which way, is not so important for me.

### Output of pd.show_versions()
<details>
  <summary>Details</summary>

```
INSTALLED VERSIONS  
------------------  
commit           : 67a3d4241ab84419856b84fc3ebc9abcbe66c6b3
python           : 3.8.5.final.0
python-bits      : 64
OS               : Darwin
OS-release       : 20.1.0
Version          : Darwin Kernel Version 20.1.0: Sat Oct 31 00:07:11 PDT 2020; root:xnu-7195.50.7~2/RELEASE_X86_64
machine          : x86_64
processor        : i386
byteorder        : little
LC_ALL           : None
LANG             : None
LOCALE           : None.UTF-8
pandas           : 1.1.4
numpy            : 1.19.4
pytz             : 2020.4
dateutil         : 2.8.1
pip              : 20.2.4
setuptools       : 50.3.1.post20201107
Cython           : None
pytest           : None
hypothesis       : None
sphinx           : None
blosc            : None
feather          : None
xlsxwriter       : None
lxml.etree       : None
html5lib         : None
pymysql          : None
psycopg2         : None
jinja2           : None
IPython          : None
pandas_datareader: None
bs4              : None
bottleneck       : None
fsspec           : None
fastparquet      : None
gcsfs            : None
matplotlib       : None
numexpr          : None
odfpy            : None
openpyxl         : None
pandas_gbq       : None
pyarrow          : None
pytables         : None
pyxlsb           : None
s3fs             : None
scipy            : None
sqlalchemy       : None
tables           : None
tabulate         : None
xarray           : None
xlrd             : None
xlwt             : None
numba            : None
```

</details>