BUG: to_json() swallows attributes of dataclasses
#### Code Sample, a copy-pastable example

```python
from dataclasses import dataclass
import pandas as pd

@dataclass()
class MyObject:
    a: "str"
    b: "str"
    c: "str"

@dataclass()
class MyObject2:
    a: "str"
    _b: "str"
    c: "str"
        

o1 = MyObject(a='a', b='b', c='c')
o2 = MyObject2(a='a', _b='b', c='c')


df = pd.DataFrame({"objects":[o1, o2]})

print(df.to_json())
print(pd._libs.json.dumps(o1))
print(pd._libs.json.dumps(o2))
```

Output:
```
{"objects":{"0":{"a":"a","c":"c"},"1":{"a":"a"}}}
{"a":"a","c":"c"}
{"a":"a"}
```

#### Problem description

While the dataclasses have three attributes depending on the naming only one or two show up in the JSON dump.

#### Expected Output

```
{"objects":{"0":{"a":"a","b"="b","c":"c"},"1":{"a":"a","_b"="b","c":"c"}}}
{"a":"a","b"="b","c":"c"}
{"a":"a","_b"="b","c":"c"}
```

#### Output of ``pd.show_versions()``

<details>

INSTALLED VERSIONS
------------------
commit           : c7f7443c1bad8262358114d5e88cd9c8a308e8aa
python           : 3.9.1.final.0
python-bits      : 64
OS               : Darwin
OS-release       : 20.5.0
Version          : Darwin Kernel Version 20.5.0: Sat May  8 05:10:33 PDT 2021; root:xnu-7195.121.3~9/RELEASE_X86_64
machine          : x86_64
processor        : i386
byteorder        : little
LC_ALL           : None
LANG             : None
LOCALE           : None.UTF-8

pandas           : 1.3.1
numpy            : 1.21.1
pytz             : 2021.1
dateutil         : 2.8.2
pip              : 21.0.1
setuptools       : 49.2.1
Cython           : None
pytest           : 6.2.4
hypothesis       : None
sphinx           : None
blosc            : None
feather          : None
xlsxwriter       : None
lxml.etree       : None
html5lib         : None
pymysql          : None
psycopg2         : 2.9.1 (dt dec pq3 ext lo64)
jinja2           : 2.11.3
IPython          : 7.20.0
pandas_datareader: None
bs4              : 4.9.3
bottleneck       : 1.3.2
fsspec           : 2021.07.0
fastparquet      : None
gcsfs            : 2021.07.0
matplotlib       : 3.3.4
numexpr          : 2.7.3
odfpy            : None
openpyxl         : 3.0.7
pandas_gbq       : 0.15.0
pyarrow          : 4.0.1
pyxlsb           : None
s3fs             : None
scipy            : 1.7.0
sqlalchemy       : 1.4.22
tables           : None
tabulate         : 0.8.9
xarray           : None
xlrd             : 1.2.0
xlwt             : None
numba            : 0.51.2
</details>
