The problem is that you are using the datetime accessor (.dt) with the property time and then you are not able to subtract the two columns from eachother. So, just leave out .dt.time and it should work.
Here is some data with 2 columns of strings
df = pd.DataFrame()
df['time_entry'] = ['12:01:00', '15:03:00', '16:43:00', '14:11:00']
df['time_entry2'] = ['13:03:00', '14:04:00', '19:23:00', '18:12:00']
print(df)
  time_entry time_entry2
0   12:01:00    13:03:00
1   15:03:00    14:04:00
2   16:43:00    19:23:00
3   14:11:00    18:12:00
Convert both columns to datetime dtype
df['time_entry'] = pd.to_datetime(df['time_entry'], format= '%H:%M:%S', errors='ignore')
df['time_entry2'] = pd.to_datetime(df['time_entry2'], format= '%H:%M:%S', errors='ignore')
print(df)
           time_entry         time_entry2
0 1900-01-01 12:01:00 1900-01-01 13:03:00
1 1900-01-01 15:03:00 1900-01-01 14:04:00
2 1900-01-01 16:43:00 1900-01-01 19:23:00
3 1900-01-01 14:11:00 1900-01-01 18:12:00
print(df.dtypes)
time_entry     datetime64[ns]
time_entry2    datetime64[ns]
dtype: object
(Optional) Specify timezone
df['time_entry'] = df['time_entry'].dt.tz_localize('US/Central')
df['time_entry2'] = df['time_entry2'].dt.tz_localize('US/Central')
Now perform the time difference (subtraction) between the 2 columns and get the time difference in number of days (as a float)
df['Diff_days1'] = (df['time_entry'] - df['time_entry2']).dt.total_seconds()/60/60/24
df['Diff_days2'] = (df['time_entry'] - df['time_entry2']) / np.timedelta64(1, 'D')
df['Diff_days3'] = (df['time_entry'].sub(df['time_entry2'])).dt.total_seconds()/60/60/24
print(df)
           time_entry         time_entry2  Diff_days1  Diff_days2  Diff_days3
0 1900-01-01 12:01:00 1900-01-01 13:03:00   -0.043056   -0.043056   -0.043056
1 1900-01-01 15:03:00 1900-01-01 14:04:00    0.040972    0.040972    0.040972
2 1900-01-01 16:43:00 1900-01-01 19:23:00   -0.111111   -0.111111   -0.111111
3 1900-01-01 14:11:00 1900-01-01 18:12:00   -0.167361   -0.167361   -0.167361
EDIT
If you're trying to access datetime attributes, then you can do so by using the time_entry column directly (not the time difference column). Here's an example
df['day1'] = df['time_entry'].dt.day
df['time1'] = df['time_entry'].dt.time
df['minute1'] = df['time_entry'].dt.minute
df['dayofweek1'] = df['time_entry'].dt.weekday
df['day2'] = df['time_entry2'].dt.day
df['time2'] = df['time_entry2'].dt.time
df['minute2'] = df['time_entry2'].dt.minute
df['dayofweek2'] = df['time_entry2'].dt.weekday
print(df[['day1', 'time1', 'minute1', 'dayofweek1',
        'day2', 'time2', 'minute2', 'dayofweek2']])
   day1     time1  minute1  dayofweek1  day2     time2  minute2  dayofweek2
0     1  12:01:00        1           0     1  13:03:00        3           0
1     1  15:03:00        3           0     1  14:04:00        4           0
2     1  16:43:00       43           0     1  19:23:00       23           0
3     1  14:11:00       11           0     1  18:12:00       12           0